Critical Multipliers in Variational Systems via Second-order Generalized Differentiation

TL;DR

This paper introduces the concepts of critical and noncritical multipliers in variational systems, providing characterizations and conditions for stability and convergence in optimization algorithms using second-order generalized differentiation.

Contribution

It extends the notions of critical multipliers to a broad variational framework and offers complete characterizations using second-order subdifferential theory.

Findings

Noncritical multipliers are equivalent to error bounds in perturbed systems.

Critical multipliers can be eliminated through full stability conditions.

Lipschitz-like properties imply robust isolated calmness of solution maps.

Abstract

In this paper we introduce the notions of critical and noncritical multipliers for subdifferential variational systems extending to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain error bound for a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.